Sulem, C.; Sulem, Pl. The well-posedness of two-dimensional ideal flow. (English) Zbl 0561.76032 J. Méc. Théor. Appl. 1983, Suppl., 217-242 (1983). This is a review of mathematical results concerning well-posedness of the Euler equation for an ideal fluid in two dimensions. When the vorticity is initially bounded (in Hölder norm), there exists for all time a unique solution which remains as smooth as the initial data. This condition is not satisfied in the context of the Kelvin Helmholtz instability where the initial velocity is discontinuous through a vortex sheet. In the latter case, existence for a short time of the vortex sheet is insured only if the sheet and the vorticity density are initially analytic. Asymptotic analysis [D. W. Moore, Proc. R. Soc. Lond., Ser. A 365, 105-119 (1979; Zbl 0404.76040)] and numerical simulations [D. I. Meiron, G. R. Baker and S. A. Orszag, J. Fluid Mech. 114, 283-298 (1982; Zbl 0476.76031)] suggest that a singularity in the vortex sheet develops in a finite time. Existence of the flow all time is then ensured only in a weak sense. The emphasis is on an elementary presentation of the methods. We have tried to bring out the hard cores of the proofs (mainly a priori estimates), leaving out some technical details. Cited in 6 Documents MSC: 76B47 Vortex flows for incompressible inviscid fluids 35Q99 Partial differential equations of mathematical physics and other areas of application Keywords:two dimensional flow; well-posedness of the Euler equation; ideal fluid; Hölder norm; Kelvin Helmholtz instability; vortex sheet Citations:Zbl 0404.76040; Zbl 0476.76031 PDFBibTeX XMLCite \textit{C. Sulem} and \textit{Pl. Sulem}, J. Méc. Théor. Appl. 1983, 217--242 (1983; Zbl 0561.76032)